\usepackage{manfnt}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{bbm}
\usepackage{slashed}
\usepackage{hyperref}

\def\identity\mathbbm{1}

\setlength{\parskip}{0.1in}

\author{Alex Nelson}
\date{\today}



\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\newtheorem{ex}{Example}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{axm}{Axiom}[section]
\newtheorem{rmk}{Remark}[section]
\newtheorem{cor}{Corollary}[section]


\def\tr{\operatorname{Tr}}
\def\vect{\operatorname{Vect}}
\def\Vect{\operatorname{Vect}}

\numberwithin{equation}{section}


%%
% This macro defines the ``dbar'' variable
%%
\def\dbar{{\textrm{\dh}}}


%%
% This macro header is what controls the ``dangerous bend'' paragraph
%%
\def\rd{\noindent\begingroup\hangindent=2pc\hangafter=-2\def\par{\endgraf\endgroup}\hbox to0pt{\hskip-\hangindent\dbend\hfill}\ignorespaces}
%%
% This command allows you to write stuff in small font size and use the
% bourbaki ``dangerous bend'' so it's great when you want to ramble on 
% about some extra stuff!
%%
\newcommand{\danger}[1] {\rd{\small {#1}}}
%%
% Don't forget that \alpha_E^{-1} = 137.035999665008
% so \alpha_{S} = 16\alpha_E = 16/137.035999665008
%%


%%
% This macro header is what controls the ``dangerous bend'' paragraph
%%
\def\ddbend{\dbend\kern1pt\dbend}

\def\rdd{\noindent\begingroup\hangindent=3pc\hangafter=-2\def\par{\endgraf\endgroup}\hbox to0pt{\hskip-\hangindent\ddbend\hfill}\ignorespaces}

\newcommand{\ddanger}[1] {\rdd{\small {#1}}}

\setlength{\marginparwidth}{1.2in}
\let\oldmarginpar\marginpar
\renewcommand\marginpar[1]{\-\oldmarginpar[\raggedleft\footnotesize #1]%
{\raggedright\footnotesize #1}}

\newcommand{\notetoself}[1] {$\langle${\it Note to self: {#1}}$\rangle$}
\newcommand{\snote}[1] {$\langle${\it \small{#1}}$\rangle$}
%%
% This macro defines the CG coefficients
%%

% Example usuage:
% \cg{m_1}{j_1}{m_2}{j_2}{M}{J}
% produces:
% < m_1 & m_2 | M 
%   j_1 & j_2 | J>

\newcommand{\cg}[6]{
  \left<
  \begin{array}{cc|c}
  #1 & #3 & #5 \\
  #2 & #4 & #6
  \end{array}
  \right>
}

%%
% This macro defines time ordering symbol
%%

\newcommand{\T}[1]{ T\left\{#1\right\} }

%%
% This macro defines the bra-ket notation
%%
%
% These are fine for ``small'' states
%
\def\bra#1{\mathinner{\langle{#1}|}}
\def\ket#1{\mathinner{|{#1}\rangle}}
%
% This braket command is fine when we are taking the inner product of only
% two things; for the expectation value of something use Braket
%
\newcommand{\braket}[2]{\langle #1|#2\rangle}
%
% The Bra Kets here are for large vectors
%
\def\Bra#1{\left<#1\right|}
\def\Ket#1{\left|#1\right>}
%
% The \Braket command is when we have <a|operator|b>
%
\newcommand{\Braket}[3]{\left<#1\right|#2\left|#3\right>}
%
% This is used for the expectation value of something, i.e. <a|operator|a>
%
\newcommand{\expect}[1]{\left<#1\right>}

\newcommand{\vacuum}[1]{\Braket{0}{#1}{0}}
